Overall, Brouwer fixed point theorem and Kakutani fixed theorem are non-constructive. Is there any established paper that demonstrates that there exists constructive proofs that do exactly what these theorems do?
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Yes, there are constructive proofs for fixed point theorems including Brouwer. Also, the proof of the Banach fixed point theorem with which I am most familiar is constructive. In fact, here is a paper all about constructive methods for fixed point theorems: